Related papers: Limit $F$-signature functions of two-variable bino…

The $F$-signature of a local ring of prime characteristic is a numerical invariant that detects many interesting properties. For example, this invariant detects (non)singularity and strong $F$-regularity. However, it is very difficult to…

Let $f$ be a diagonal hypersurface in $A_p=\mathbb{F}_p[[x_1,\dots,x_n]]$. We study the behavior of the function $\phi_{f,p}({a}/{p^e})=p^{-ne}\dim_{\mathbb{F}_p}\big(A_p/(x_1^{p^e},\dots,x_n^{p^e},f^a)\big)$ which encodes information about…

We study the shape of the F-signature function of a $d$-dimensional quotient singularity $\mathbb{k}[[ x_1,\ldots,x_d]]^G$, and we show that it is a quasi-polynomial. We prove that the second coefficient is always zero and we describe the…

We define the dual F-signature of modules, which is equivalent to the F-signature if the module is the base ring. By using this invariant, We give characterizations of regular, F-regular, F-rational, and Gorenstein singularities.

F-signature is an important numeric invariant of singularities in positive characteristic that can be used to detect strong F-regularity. One would like to have a variant that rather detects F-rationality, and there are two theories that…

This paper establishes uniform bounds in characteristic $p$ rings which are either F-finite or essentially of finite type over an excellent local ring. These uniform bounds are then used to show that the Hilbert-Kunz length functions and…

The $F$-signature is a numerical invariant defined by the number of free direct summands in the Frobenius push-forward, and it measures singularities in positive characteristic. It can be generalized by focussing on the number of non-free…

Hilbert-Kunz multiplicity and F-signature are numerical invariants of commutative rings in positive characteristic that measure severity of singularities: for a regular ring both invariants are equal to one and the converse holds under mild…

This paper focuses on a numerical invariant for local rings of characteristic $p$ called $h$-function, that recovers several important invariants, including the Hilbert-Kunz multiplicity, $F$-signature, $F$-threshold, and $F$-signature of…

We study a conjecture of Carvajal-Rojas, Schwede and Tucker which states that for a complex KLT singularity $(R, \mathfrak{m})$, the F-signatures of the reductions of $R$ to characteristic $p \gg 0$ remain bounded away from zero as $p \to…

For any fixed globally F-regular projective variety X over an algebraically closed field of positive characteristic, we study the F-signature of section rings of X with respect to the ample Cartier divisors on X. In particular, we define an…

This paper studies properties of certain hypersurfaces in prime characteristic: we give a sufficient and necessary conditions for some classes of such hypersurfaces to have Finite $F$-representation Type (FFRT) and we compute the…

The dual $F$-signature is a numerical invariant defined via the Frobenius morphism in positive characteristic. It is known that the dual $F$-signature characterizes some singularities. However, the value of the dual $F$-signature is not…

We study $F$-signature under proper birational morphisms $\pi : Y \to X$, showing that $F$-signature strictly increases for small morphisms or if $ K_Y \geq \pi ^*K_X$. In certain cases, we can even show that the $F$-signature of $Y$ is at…

Suppose R is a Noetherian local ring with prime characteristic p>0. In this article, we show the existence of a local numerical invariant, called the F-signature, which roughly characterizes the asymptotic growth of the number of splittings…

We express the F-signature of the coordinate ring of an affine toric variety as the volume of a certain polytope, generalizing a formula of Watanabe and Yoshida. We also compute the F-signature of pairs and triples of a toric singularity.

We extend the theory of $p$-fractals of Monsky and Teixeira by introducing the notion of weak $p$-fractal. We prove that for a hypersurface $f$ having rational Hilbert-Kunz series is equivalent to the weak $p$-fractality of the associated…

We study the differential symmetric signature, an invariant of rings of finite type over a field, introduced in a previous work by the authors in an attempt to find a characteristic-free analogue of the F-signature. We compute the…

We provide a family of examples where the $F$-pure threshold and the log canonical threshold of a polynomial are different, but where $p$ does not divide the denominator of the $F$-pure threshold (compare with an example of…

We show that the intersection of the irreducible components of a hypersurface defined by a polynomial with square-free support has F-rational singularities in characteristic $p>0$. As a consequence, we obtain that hypersurfaces defined by…